

A007615


Primes with unique period length (the periods are given in A007498).
(Formerly M2890)


7



3, 11, 37, 101, 333667, 9091, 9901, 909091, 1111111111111111111, 11111111111111111111111, 99990001, 999999000001, 909090909090909091, 900900900900990990990991, 9999999900000001, 909090909090909090909090909091, 900900900900900900900900900900990990990990990990990990990991
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OFFSET

1,1


COMMENTS

Additional terms are Phi(n,10)/gcd(n,Phi(n,10)) for the n in A007498, where Phi(n,10) is the nth cyclotomic polynomial evaluated at 10.


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 2224.


LINKS

Ray Chandler, Table of n, a(n) for n = 1..31 (first 25 terms from T. D. Noe)
Index entries for sequences related to decimal expansion of 1/n
C. K. Caldwell, The Prime Glossary, unique prime
Makoto Kamada, Factorizations of Phi_n(10)


FORMULA

a(n) = A061075(A007498(n)) [From Max Alekseyev, Oct 16 2010]
a(n) = A006530(A019328(A007498(n))).  Ray Chandler, May 10 2017


EXAMPLE

3 is the only prime p such that decimal expansion of 1/p has (nontrivial) period exactly 1.


MATHEMATICA

nmax = 50; periods = Reap[ Do[ p = Cyclotomic[n, 10] / GCD[n, Cyclotomic[n, 10]]; If[ PrimeQ[p], Sow[n]], {n, 1, nmax}]][[2, 1]]; Cyclotomic[#, 10] / GCD[#, Cyclotomic[#, 10]]& /@ periods // Prepend[#, 3]& (* JeanFrançois Alcover, Mar 28 2013 *)


CROSSREFS

Cf. A007498, A040017, A002371, A048595, A006883, A007732, A051626, A061075, A006530, A019328.
Sequence in context: A061075 A005422 A040017 * A065540 A084171 A192875
Adjacent sequences: A007612 A007613 A007614 * A007616 A007617 A007618


KEYWORD

nonn,nice,easy,base


AUTHOR

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein


STATUS

approved



